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Wasserstein projections in the convex order were first considered in theframework of weak optimal transport, and found application in various problemssuch as concentration inequalities and martingale optimal transport. Indimension one, it is well-known that the set of probability measures with agiven mean is a lattice w.r.t. the convex order. Our main result is that,contrary to the minimum and maximum in the convex order, the Wassersteinprojections are Lipschitz continuity w.r.t. the Wasserstein distance indimension one. Moreover, we provide examples that show sharpness of theobtained bounds for the 1-Wasserstein distance.

Lipschitz continuity of the Wasserstein projections in the convex order on the line